The Adams-Novikov spectral sequence and the Landweber exact functor
theorem
Gabriel Valenzuela
April 22, 2013
1
The Adams-Novikov spectral sequence
If E is a sufficiently nice ring spectrum, there’s a spectral sequence
b
E2 = Exts,t
E∗ E (π∗ E, E∗ X) → πt−s (X),
b is the E-nilpotent completion of X, and Exts is the sth right derived functor of HomE E in
where X
∗
b is defined in Bousfield’s 1979 paper
the category of comodules over the Hopf algebroid (π∗ E, E∗ E). X
on localization of spectra: it’s an E∗ -local object receiving a map from X, which in nice cases is an E∗ b
isomorphism, forcing LE X ' X.
We’ll be most interested in the cases E = HZ/p, M U , and BP , the third being the ANSS. All of these
have nice properties, such as flatness of the associated Hopf algebroid, connectivity, . . . .
Theorem 1 (Novikov, 1967). For any spectrum X, there’s a natural spectral sequence Ers,t , with dr : Ers,t →
Ers+r,t+r−1 with
(a) E2s,t ∼
= ExtBP∗ BP (BP∗ , BP∗ X);
(b) If X is connective, then E2s,t ⇒ πt−s (X(p) ).
The advantage of the ANSS over the classical ASS is that BP∗ BP is concentrated in degrees divisible by
2(p − 1). Thus, for example, if BP∗ X is also concentrated in such degrees, then Ers,t = 0 unless 2(p − 1)|t,
and so there are fewer differentials. This phenomenon is called sparseness.
In order to compute the ANSS E2 page, we need to understand the Hopf algebroid (BP∗ , BP∗ BP ).
Recall that a Hopf algebroid (A, Γ) over a commutative ring K is a cogroupoid object in the category of
(commutative) K-algebras. Thus, for any ring B, Hom(A, B) and Hom(Γ, B) are naturally respectively the
objects and morphisms of a groupoid.
If E∗ E is flat over E∗ (we say that ‘E is flat’ for short), then (E∗ , E∗ E) is a Hopf algebroid with structure
maps
ηL = π∗ (E ∧ S → E ∧ E)
(source)
ηR = π∗ (S ∧ E → E ∧ E)
(target)
= π∗ (E ∧ E → E)
(identity)
flip
c = π∗ (E ∧ E → E ∧ E)
(inverse)
∆ = π∗ (E ∧ S ∧ E → E ∧ E ∧ E)
(composition).
Flatness is used to define the last map, which we should be able to identify as a map E∗ E → E∗ E ⊗E∗ E E∗ E.
Definition 2. A (left) Γ-comodule M is a left A-module with a coaction map
ψ : M → Γ ⊗A M
which is counital and coassociative.
1
2
For example, if X is a spectrum and E is as above, then E∗ X is an E∗ E-comodule with ψ induced by
E ∧ S ∧ X → E ∧ E ∧ X.
The structure of (BP∗ , BP∗ BP ) (and likewise (M U∗ , M U∗ M U )) comes naturally from formal groups.
Specifically, let F GL : CRing → Set and SI : CRing → Set send a ring R respectively to its set of formal
group laws and its set of strict isomorphisms of formal group laws. These are the objects and morphisms of
a groupoid in the obvious way. Now, F GL = Hom(L, ·) where L = Z[x1 , x2 , · · · ] with |xi | = 2i (the Lazard
ring), and SI = Hom(LB, ·) where LB = L ⊗ Z[b1 , b2 , ·] with |bi | = 2i. By Quillen’s theorem, these are
isomorphic to M U∗ and M U∗ M U respectively.
Theorem 3 (Landweber 1967, Novikov 1967, independently!). These maps give an isomorphism of Hopf
algebroids (L, LB) → (M U∗ , M U∗ M U ).
∼
=
Likewise, we have (V, V T ) → (BP∗ , BP∗ BP ), where V = Z(p) [v1 , v2 , . . . ] with |vi | = 2(pi − 1), V T =
V ⊗ Z(p) [t1 , t2 , . . . ] with |ti | = 2(pi − 1), and V and V T respectively corepresent the p-typical formal group
laws functor and the strict isomorphisms of p-typical formal group laws functor.
2
The Landweber exact functor theorem
If E is a complex-oriented cohomology theory with a fixed orientation class, we can construct a formal group
law over E∗ in a natural way. We now ask: when can we go back?
Let F (x, y) ∈ F GL(R∗ ) for R∗ a graded ring; this gives a map M U∗ → R∗ . Thus there’s a functor
X 7→ M U∗ X ⊗M U∗ R∗ .
This satisfies all the axioms of a homology theory with the exception of turning cofiber sequences into long
exact sequences: tensoring with R∗ has destroyed exactness. If R∗ were flat over M U∗ , we’d be fine, but
this is too strong.
The same question can be asked p-typically: if R∗ has a p-typical formal group law represented by
BP∗ → R∗ , then when is
X 7→ BP∗ X ⊗BP∗ R∗
a homology theory?
Theorem 4 (Landweber exact functor theorem, Landweber 1976). For a fixed BP∗ -module R∗ , the above
functor is a homology theory iff for all n, the sequence (p, v1 , v2 , . . . , vn ) is a regular sequence in R∗ (that is,
each vn is a non-zero-divisor in R∗ /(p, v1 , . . . , vn−1 )).
The proof is via the study of the category of (finitely presented) BP∗ BP -comodules. One arrives at the
following theorems:
Theorem 5 (Landweber 1973). The only prime ideals of BP∗ which are also BP∗ BP -comodules are the
ideals In = (p, v1 , . . . , vn−1 ) for 0 ≤ n ≤ ∞.
Theorem 6 (Landweber filtration theorem, Landweber 1973). Any BP∗ BP -comodule M which is finitely
presented as a BP∗ -module has a filtration by comodules
0 = M0 ⊆ M1 ⊆ · · · ⊆ Mn = M
with Mi+1 /Mi ∼
= BP∗ /Ini .
(We need to be careful in general about our finiteness conditions when dealing with non-noetherian rings.
However, since BP∗ is coherent as a ring, a module is finitely presented iff it is coherent.)
Proof of LEFT. Consider the exact sequences
v
n
0 → BP∗ /In →
BP∗ /In → BP∗ /In+1 → 0.
The sequences given in the theorem statement are regular in R∗ iff tensoring with R∗ preserves exactness
of these sequences, which is true iff TorBP∗ (BP∗ /In , R∗ ) = 0. But by the filtration theorem, this is true iff
2. THE LANDWEBER EXACT FUNCTOR THEOREM
3
TorBP∗ (M, R∗ ) = 0 for any finitely presented BP∗ BP -comodule M , which is true in turn iff tensoring with
R∗ is exact on the category of finitely presented BP∗ BP -comodules. Finally, if X is a finite CW-complex,
then BP∗ X is finitely presented, so we establish the excision axiom on the finite CW-complexes, which
implies it for all spectra.
The only nontrivial step in the converse is showing that enough finitely presented BP∗ BP -comodules
show up as BP∗ X for finite CW-complexes X – we need some analogue of the periodicity theorem.
Example 7.
• Let Fa (x, y) = x + y over Q. Then M U∗ (X) ⊗ Q ∼
= H∗ (X; Q).
• Let Fm (x, y) = x + y + βxy over Z[β, β −1 ] with |β| = 2. Then M U∗ (X) ⊗ Z[β, β −1 ] ∼
= K∗ X.
• For every elliptic curve, there’s a natural FGL over a ring representing modular forms with certain
Fourier coefficients, and this gives a homology theory Ell∗ (X), called elliptic homology.
• The Johnson-Wilson theories E(n), with E(n)∗ = Z(p) [v1 , . . . , vn , vn−1 ], are constructed via the LEFT,
and we have to invert vn in order to achieve Landweber exactness.
• Importantly, Morava K-theory K(n) cannot be built in this way. However, there’s a spectral sequence E2 = TorBP∗ (BP∗ X, K(n)∗ ) ⇒ K(n)∗ (X) which arises directly from the failure of Landweber
exactness. (It’s just a Künneth spectral sequence.)